Properties of Estimators

1
Properties of Estimators
Parameters
Describe the population
Statistics
Describe samples.
But we use it to estimate population parameters

1. Sufficiency
2. Un-biased
3. Resistance
4. Efficiency

2
Sample Variance as an Unbiased estimator
UNbiased
Biased
Example population
y 1, 2, 3
3
Samples of Two from the above population
Sample y 1, 2
If
If
4
n versus n-1 All permutations
Sample mean Var(n) Var(n-1)
1,1 1.0 0 0
1,2 1.5 .25 .50
1,3 2.0 1.0 2.0
2,1 1.5 .25 .50
2,2 2.0 0 0
2,3 2.5 .25 .50
3,1 2.0 1.0 2.0
3,2 2.5 .25 .50
3,3 3.0 0 0
E 2.0 .333 .667
5
E expected value
Remember
Not all n-1 estimates are better than their n
counterparts. But, on average, n-1 is superior
(Unbiased).
Degrees of Freedom